THEORY AND METHODS
(Gauss-Elimination Method)
History of Gauss-Elimination Method
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It is also known as row reduction. The method is named after Carl Friedrich Gauss (1777–1855), although it was known to Chinese mathematicians as early as 179 CE. The method of Gaussian elimination appears in the Chinese and its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 CE, but parts of it were written as early as approximately 150 BCE. It was commented on by Liu Hui in the 3rd century.
The method in Europe stems from the notes of Isaac Newton. In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as Arithmetica Universalis in 1707 long after Newton left academic life. The notes were widely imitated, which made Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century.
Process Gauss-Elimination Method
Gaussian elimination is a systematic application ion of elementary row operations to a system of linear equations in order to convert the system to upper triangular form. Once the coefficient matrix is in upper triangular form, we use back substitution to find a solution. The general procedure for Gaussian elimination can be summarized in the following steps:
1. Write the augmented matrix for the system of linear equation.
2. Use elementary operation on {A/b} to transform A into upper triangular form. If a zero is located on the diagonal, switch the rows until a non zero is in that place. If you are unable to do so, stop; the system has either infinite or no solution.
3. Use back substitution to find the solution of the problem.
Calculating Problem Theoretically Using Gauss-Elimination Method
By analyzing the following circuit we get four equations and these are,
So that,
2I4=26
=>I4=13
3.1587 I3-3.1587 I4 = 12.5714
=>I3= 16.9799
7.875 I2- 5.5 I3-2.375 I4=2.25
=>I2=16.0653
8 I1- 3 I2- 4 I3- I4= 6
=>I1=16.8894
Matlab Code (Gauss-Elimination Method)
a = [8 -3 -4 -1 6
-3 9 -4 -2 0
-4 -4 9 -1 8
-1 -2 -1 6 12];
[m,n]=size(a);
for j=1:m-1
for z=2:m
if a(j,j)==0
t=a(j,:);a(j,:)=a(z,:);
a(z,:)=t;
end
end
for i=j+1:m
a(i,:)=a(i,:)-a(j,:)*(a(i,j)/a(j,j));
end
end
x=zeros(1,m);
for s=m:-1:1
c=0;
for k=2:m
c=c+a(s,k)*x(k);
end
x(s)=(a(s,n)-c)/a(s,s);
end
disp('Gauss elimination method:');
ax'
Matlab answers
I1= 16.8894
I2= 16.0653
I3= 16.9799
I4= 13.0000
Advantages of Using Gauss Elimination Method
• It is probably the best method for solving systems of equations without having a graphing calculator or computer program.
• Using this method, we can find
i) The rank of a matrix
ii) The determinant of a matrix
iii) The inverse of an invertible square matrix
Disadvantages of Using Gauss Elimination Method
i) In the practical case of sparse matrices that it needs way more memory and potentially more time.
ii) More than 3*3 matrix, it becomes very complex to solve the problem.
